Suleimanov, B. A.; Dyshin, O. A. Application of discrete wavelet transform to the solution of boundary value problems for quasi-linear parabolic equations. (English) Zbl 1286.65136 Appl. Math. Comput. 219, No. 12, 7036-7047 (2013). Summary: The wavelet method for solving the linear and quasi-linear parabolic equations under initial and boundary conditions is set out. By applying regular multi-resolution analysis and received formula for differentiating wavelet decompositions of functions of many variables the problem is reduced to a finite set of linear and accordingly nonlinear algebraic equations for the wavelet coefficients of the problem solution. The general scheme for finite-dimensional approximation in the regularization method is combined with the discrepancy principle. For quasi-linear parabolic equations the convergence rate of an approximate weak solution to a classical one is estimated.{ }The proposed method is used for constructing stable approximate wavelet decompositions of weak solutions to boundary value problems for the unsteady porous-medium flow equation with discontinuous coefficients and inexact data. MSC: 65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs 35K59 Quasilinear parabolic equations 35R05 PDEs with low regular coefficients and/or low regular data 76S05 Flows in porous media; filtration; seepage 65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs 65T60 Numerical methods for wavelets Keywords:weak and approximate weak solutions to initial boundary value problems; regular multi-resolution analysis; finite-dimensional approximation scheme; gradient-type iterative method; irregular operator equation; wavelet method; quasi-linear parabolic equations; regularization method; discrepancy principle; unsteady porous-medium flow equation; discontinuous coefficients Software:UNCMND; pchip PDFBibTeX XMLCite \textit{B. A. Suleimanov} and \textit{O. A. Dyshin}, Appl. Math. Comput. 219, No. 12, 7036--7047 (2013; Zbl 1286.65136) Full Text: DOI